In the last decade, we have witnessed an increased global focus on extremist behaviour. Although the most obvious causes have been significant terrorist atrocities, it should be noted that the study of extremism is not limited to only political and religious ideology. Considering sporting and celebrity fandoms, brand loyalties, and entertainment allegiances, it can be hard to imagine any aspect of life that is not influenced by the dynamics of extremist behaviour. Given its prevalence, it is perhaps surprising how little we understand the dynamics of extremism, crucial in mitigating or avoiding disaster. The purpose of my research was to provide further insight into the workings of extremist behaviour, presenting a deeper understanding of its causes and investigating what can be done to alleviate problematic situations. In particular, I was to study how extremist viewpoints can arise in otherwise moderate populations.

While many diverse areas of academia have brought their expertise to bear on these problems, the work itself belongs to its own field called Opinion Dynamics. Since its initial coining, Opinion Dynamics has come to refer to a broad class of different models that can be applied to fields ranging from sociological phenomena to pure mathematics and the study of physics. My focus however, was on the development of various models aimed at explaining the causes of extremist populaces, as well as presenting a new model building on previous work to produce a simpler but more complete understanding of these complex behaviours.

We begin by imagining a discussion in a room full of experts. In this ideal situation, each expert stands up and expresses their opinion in turn. After everyone has participated, they change their opinions in light of the views of the others. The experts repeat this process until a consensus has been reached. Provided that each expert adjusts their opinion in agreement with that of the majority of the others, a consensus will always eventually be reached [1]. Although ideal, this is hardly a realistic model of expert behaviour. To make this model a little more realistic we add another clause; this time each expert has an uncertainty about his or her own opinion. As uncertainty increases, so too does the range of acceptable opinions. With this additional rule, it is clear that now the experts will not always reach an agreement. This model is called the Bounded Confidence (BC) model [2]. A similar model was also proposed, based on the BC model, but with two changes. Firstly, to improve the realism of the model, experts would meet in randomly selected pairs and secondly, experts would be most heavily influenced by those whose opinions most closely matched their own with the lowest uncertainties. Owing to the second change, this new model was named the Relative Agreement (RA) model [3].

Although the BC and RA model populations are not guaranteed to reach a consensus, we can intuitively understand that they will at least reach a stable state, with small groups reaching agreements and stubbornly refusing to listen to anyone else. The question now becomes, what happens if we throw some extreme viewpoints into the mix? For fairness, we add an equal number of opposing extremists, and say that they are so confident that they can be highly persuasive but are not themselves easily influenced. What happens to the initially moderate experts? In the BC model, if they begin confident, they converge to a central opinion and ignore the extremists. This is called Central Convergence. Conversely if they begin uncertain, the population will split in half, tending towards opposite extremes, known as Bipolar Convergence. The RA model however, has a third, very interesting alternative. When the moderate population has a high uncertainty, the population demonstrates an instability that may lead to the majority tending towards a single extreme viewpoint, aptly named Single Extreme Convergence. This ability to demonstrate these three convergences is significant as they are particularly analogous to real world behaviour. If we consider most countries as exhibiting behaviour similar to Central Convergence, then there are obvious and costly historic examples of Bipolar and Single Extreme Convergence. To become better acquainted with these behaviours, the app in Figure 1 allows readers to experiment with the RA model by varying its parameters.

Parameters

Typical uncertainty

Proportion of Extremists

Agreement weight

Figure 1: Use the sliders on the right to adjust the parameters and click run. The position of a line on the y-axis shows the opinion of that expert, while the x-axis shows the progression of time. The colour of the line highlights the expert’s uncertainty; red showing a confident expert and green being completely unsure. As can be seen by their red colouring, the extremists are highly confident in their opinion.

It is all well and good to find examples of these convergences with particular settings, but it is more helpful to know how the population is likely to behave for any combination of parameters. First, we must produce a metric for classifying the output of a single simulation. Fortunately, a simple calculation can be made based on the proportions of the initially moderate experts that become extreme. Thus we can classify the convergence of a single simulation using a y-value, where a value of 0.0 means Central Convergence, 0.5 shows Bipolar Convergence and 1.0 is Single Extreme Convergence. By using y-values, it is possible to produce heat map graphs from billions of individual simulation runs to show how a population is likely to behave with various settings. Figure 2 shows the quintessential y-value and standard deviation patterns of the basic RA model for different proportions of initially extreme experts and how uncertain the remainder of experts are at the start of the experiment.

Figure 2: The graph on the left shows the average pattern of the y-value. Areas in white represent consistent Central Convergence. Dark yellow and orange represent Bipolar Convergence and dark red areas show consistent Single Extreme Convergence. That there are no areas that can consistently produce Single Extreme Convergence can be explained by the standard deviation graph on the right. As can be seen, areas that can lead to Single Extreme Convergence must be so unstable as to allow for a variety of different convergence types.

Although the original publication of the RA model did not produce the correct heat map graphs [4], partly due to author error and in part down to bad science, the model itself has become well-known and as such, publishing these corrected graphs was of the utmost importance.

In recent years researchers have decided that the model could be improved. Instead of allowing any expert to be paired with any other, there would be pairing restrictions based on who the expert knows. While this modification was designed to improve the realism of the model, the social networks used to illustrate the connections between experts were incredibly abstract. To rectify this, my research moved on to examining how the RA model behaves if we use more realistic social networks.

Unfortunately generating random, but realistic social networks is not a trivial problem. Examination of real world networks (like friends on Facebook) shows three main qualities:

These networks are highly clustered, meaning that it is very likely that any two of your friends will know each other.

Between any two random people in the population, the number of connections will typically be surprisingly low (consider the Six Degrees of Kevin Bacon).

There exists a power law in the number of connections that people have, that is to say, a small number of people will be connected to a surprisingly high number of people.

As it stands, it is possible to generate randomised networks that possess two of the three qualities but not all three. One of the best compromises is a tuneable algorithm that produces networks with the low average path lengths (number 2), and dependent on the value of the variable µKE, elements from number 1 and 3. If µKE is set to 0.0, then the population will be highly clustered and at 1.0 the power law will be fully observable. Figure 3 shows how the heat map graph (like that in Figure 2) changes as we alter the social network in which the RA model operates.

Figure 3: Heat map graphs showing the average y-value and the standard deviation as we change the value of µKE from 0.0 (at the top) to 1.0 (at the bottom) in increments of 0.2. As can be seen, highly clustered populations are considerably more stable than any other type, so how do highly clustered populations converge to unusual patterns in real life?

One particularly important question that has been left unanswered is where the extremists come from. The RA model shows that extremism begets extremists, but fails to answer where that initial set of extremists comes from. Curiously, the answer comes from research that has been largely ignored in this field, despite being its empirical equivalent, capable of justifying the abstract work. Social Judgement Theory [5] comes from research in psychology and scientifically validates the intuitive leaps described in the RA model. However it goes further and states that, similar to how agreeing experts will compromise on their opinions, disagreeing experts will adjust their own opinion away from their partner. As a well-known phenomenon featuring in multiple areas of psychology, it is somewhat disappointing that it has not been noticed in the computational areas of Opinion Dynamics.

Therefore I proposed a new model, based on the RA model, which included a disagreement dynamic for when paired experts strongly disagreed with each other. Named the Relative Disagreement (RD) model in honour of the RA model, there is no longer any need for initially extreme experts, nor uniform uncertainties. This simple change, allows a population to exhibit all three convergence types without the artificial need for extremists to influence the results. Figure 4, like Figure 1, allows users to experiment with the parameters in the RD model to see how they affect the outcome.

Parameters

Probability of disagreement

Agreement weight

Disagreement weight

Figure 4: Use the sliders on the right to adjust the parameters and click run. The position of a line on the y-axis shows the opinion of that expert, while the x-axis shows the progression of time. The colour of the line highlights the expert’s uncertainty; red showing a confident expert and green being completely unsure. As can be seen by their red colouring, the extremists are highly confident in their opinion.

Although the variable parameters from Figures 2 and 3 no longer exist in the RD model, we can control the values of the influence that agreements and disagreements have on experts to produce a new type of heat map graph, like that shown in Figure 5.

Figure 5: Heat map graphs showing the average y-value and the standard deviation for the RD model. µRA is the weight applied when experts agree and µRD is the influence for a disagreement. Note that because we have used different parameters, this graph is not strictly comparable with Figure 2. It is noticeable that the disagreement influence is a more important value in producing instability in the population.

To provide further validation for this new RD model, as well as improving the realism of the model, it was necessary to consider the realistic social networks as before. Figure 6 shows the y-value pattern change as the value of µKE changes from highly clustered towards the power law.

Figure 6: Heat map graphs showing the average y-value and the standard deviation as we change the value of µKE from 0.0 (at the top) to 1.0 (at the bottom) in increments of 0.2. With the RD model, clustering the population shows a greatly diminished effect on stabilising the population. This is because in the RA model the scope of extremist influence was inhibited, but with the RD model, all population members can be the cause of extremism.

As is highlighted by Figure 6, the RD model explains with greater simplicity and elegance how extremist viewpoints can become the mainstream in an otherwise moderate (and clustered) population. As it is based on theory founded on empirical evidence, it can also be thought of as one of the most accurate explanations for these group-wide behaviours.

Given that these sociological phenomena have been linked to serious humanitarian disasters it is crucial that we develop a solid understanding to enable us to better prevent future crises. Furthermore, this dynamic can be applied to almost every area of our lives and learning how to influence these changes will be a central theme of research for many years to come.